In pulsed neutron decay logging, a logging tool contains a small neutron generator capable of being turned on and off in a controlled manner to produce bursts of high-energy neutrons, one or more radiation detectors to detect secondary radiation resulting from nuclear interactions between the neutrons emitted from the generator and nuclei comprising the materials in the borehole region and the formation, circuitry to record the detection times relative to a time reference related to the neutron production bursts, downhole and uphole circuitry to control operation of the tool, and means (generally uphole) to process the spectrum of detection times (commonly called the die-away or decay spectrum) to estimate earth formation properties useful in the evaluation of oil and gas reservoirs. The logging tool is passed through a borehole (uncased or cased and with or without tubing depending on the application), measurements are made as a function of depth, and a log of the results of the data processing is recorded as a function of depth.
The detector system can be designed to detect gamma rays resulting from capture of thermal neutrons produced as a result of slowing down and thermalization of the high-energy source neutrons or the thermal neutrons themselves. Since most commercial pulsed neutron decay systems presently in use are based on pulsed neutron capture measurements, the following background discussion will be focused on such measurements.
Two references providing background information on pulsed neutron capture logging are U.S. Pat. No. 4,409,481 to Smith, Jr. et al., and U.S. Pat. No. 4,926,044 to Wraight. These references, taken together, give a balanced view of the history of developments in pulsed neutron capture logging. Some of the highlights are discussed below.
Initially, the decay process was modeled by the simple expression: EQU N(t)=N.sub.o e.sup.-t/.tau., (1)
where N.sub.o is the measured capture gamma-ray intensity in a single detector at time 0, N(t) is the intensity at time t, and .tau. is the mean lifetime of thermal neutrons in the formation. Counting rates N.sub.1 and N.sub.2 were measured in two time gates of equal duration beginning at times t.sub.1 and t.sub.2, and .tau. was determined from the expression: ##EQU1## The times t.sub.1 and t.sub.2 were assumed to be set sufficiently long after the end of the neutron burst that any early deviations from single-exponential decay were negligible for the counting gates. It was also assumed that the intrinsic (i.e., true) formation capture cross section could be obtained from the relation: ##EQU2## where v is the average thermal neutron speed.
As time went on, experience began to show that there were two significant problems arising from the simple approach described above:
i) the single-exponential model is not an adequate practical representation of the die-away of capture gamma rays in the borehole environment; and PA1 ii) the true formation capture cross section cannot generally be obtained as simply as the relation above.
The first problem was addressed by using a two-exponential model: EQU N(t)=N.sub.b e.sup.-t/.tau..sbsp.b +N.sub.f e.sup.-t/.tau..sbsp.f( 4)
Measurements were made of essentially all of the statistically meaningful die-away data and they were fit to the above expression to obtain .tau..sub.b and .tau..sub.f. The shorter of the two lifetimes (assumed here to be .tau..sub.b) was taken to be characteristic of the borehole materials, while the longer one (.tau..sub.f) characterized the formation. The second problem was addressed by recognizing that .tau. (for a single-exponential model), .tau..sub.b, and .tau..sub.f were dependent on effects of thermal neutron diffusion. Therefore, one could write ##EQU3## and develop empirical diffusion corrections to be applied to .SIGMA..sub.b and .SIGMA..sub.f to obtain the intrinsic cross sections. An additional correction that must be applied is based on the fact that, because of diffusion, the values of .tau..sub.b and .tau..sub.f are not independent of each other. Thus, even if the borehole and formation have the same diffusion properties, a correction is needed to obtain the intrinsic values of borehole and formation capture cross sections.
There are practical difficulties in determining the parameters of a two-exponential model from the observed data. Least-squares fitting has been used, but unless it is modified in some way it is very slow computationally for application in the field where it is desirable to analyze the data in real time. Successive stripping has been proposed, wherein the two decay components are separately estimated and are alternately stripped mathematically from the observed data, followed by reestimation and stripping until some convergence criterion is satisfied. A combination of successive stripping and least-squares fitting has been used successfully by Halliburton (see the aforementioned U.S. Pat. No. 4,409,481).
Schlumberger has proposed and implemented a method to build diffusion effects into the die-away model based on a modification of the two-exponential model: EQU N(t)=N.sub.b t.sup.-.gamma..spsb.b e.sup.-t/.tau..spsb.b +N.sub.f t.sup.-.gamma..spsb.f e.sup.-t/.tau..spsb.f (b 7)
A reference to D. K. Steinman, et al., "Dual-Burst Thermal Decay Time Logging Principles", SPE Formation Evaluation, vol. 3, no. 2, pp. 377-385, June 1988. The factors t.sup.-.gamma..spsb.b and t.sup.-.gamma..spsb.f, where .gamma..sub.b and .gamma..sub.f are empirically determined, are claimed to account for diffusion effects. Recent comparative studies by Shell indicate that there is little, if any, significant improvement to be gained in determining intrinsic cross sections by using the above diffusion-modified decay model (F. G. van den Berg, "The Capability of Pulsed Neutron Capture Logging to Determine Oil and Gas Saturations", SPE Paper 19614, 64th Annual Technical Conference and Exhibition of SPE, San Antonio, October 1989; and R. J. M. Bonnie, "Evaluation of Various Pulsed Neutron Capture Logging Tools Under Well-Defined Laboratory Conditions", The Log Analyst, vol. 35, no. 2, pp. 46-51, March-April 1994).
The objects of the present invention are to (i) provide novel means for estimating model parameters from observed die-away data in a numerically stable and computationally efficient manner; (ii) provide novel means for obtaining intrinsic values of the macroscopic thermal neutron absorption cross section of the formation and the materials in the borehole region; and (iii) provide novel means for determining formation hydrogen index, from which porosity may be estimated, based on the parameters characterizing a two-exponential decay model.